Immediately we see a problem. The expression ln 0 is undefined. The function ln x approaches negative infinity when x approaches zero from the right side (that is x > 0 and calculating ln x as x is decreasing).

However, when x approaches from the left side, ln x approaches -infinity + i * π. There is no complex number makes the statement A^x = 0 true.

Therefore, A^x has no solutions. There is no x that makes statement (IV) in this section true.

A^x + B^x = 0

Let's see if we can get a closed form solution for x.

(I) A^x + B^x = 0

Assume that B can be rewritten as B = A^n., where n is any power. Then:

(II) A^x + (A^n)^x = 0(III) A^x + (A^x)^n = 0

Factoring out A^x:

(IV) A^x * (1 + (A^x)^(n-1) ) = 0

From (IV) either:A^x = 0, which can not happen sense from the previous section, A^x has no roots, real or complex.

So we have to look at the other possibility: (1 + (A^x)^(n-1) ) = 0.

Then:(V) (A^x)^(n-1) = -1(VI) A^x = -1^(1/(n-1))

The polar representation for -1 + 0i is 1*e^(i*π), a general solution for (VI) is: